General Congruences Modulo 5 and 7 for Colour Partitions

TL;DR

This paper establishes general congruences modulo 5 and 7 for the colour partition function $p_r(n)$, expanding understanding of partition functions with colored parts through $q$-series identities inspired by Ramanujan.

Contribution

It introduces broad congruences for $p_r(n)$ for general $r$, extending previous specific case studies using $q$-series techniques.

Findings

Proves congruences modulo 5 and 7 for $p_r(n)$ for general $r$

Utilizes $q$-series identities in the proof

Generalizes known results for specific $r$ values

Abstract

For any positive integers $n$ and $r$, let $p_r(n)$ denotes the number of partitions of $n$ where each part has $r$ distinct colours. Many authors studied the partition function $p_r(n)$ for particular values of $r$. In this paper, we prove some general congruences modulo $5$ and $7$ for the colour partition function $p_r(n)$ by considering some general values of $r$. To prove the congruences we employ some $q$-series identities which is also in the spirit of Ramanujan.